Given a subgroup $H$ of Group $G$, the left cosets of $H$ in $G$ are sets of the form $\{ gh : h \in H \}$, for some $g \in G$. This is written $gH$ as a shorthand.

Similarly, the right cosets are the sets of the form $Hg = \{ hg: h \in H \}$.

Examples

%%%knows-requisite(Symmetric group):

Symmetric group

In $S_3$, the Symmetric group on three elements, we can list the elements as $\{ e, (123), (132), (12), (13), (23) \}$, using cycle notation. Define $A_3$ (which happens to have a name: the Alternating group) to be the subgroup with elements $\{ e, (123), (132) \}$.

Then the coset $(12) A_3$ has elements $\{ (12), (12)(123), (12)(132) \}$, which is simplified to $\{ (12), (23), (13) \}$.

The coset $(123)A_3$ is simply $A_3$, because $A_3$ is a subgroup so is closed under the group operation. $(123)$ is already in $A_3$. %%%

[todo: more examples, with different requirements]

Properties

• The left cosets of $H$ in $G$ [set_partition partition] $G$. (Proof.)
• For any pair of left cosets of $H$, there is a bijection between them; that is, all the cosets are all the same size. (Proof.)

Why are we interested in cosets?

Under certain conditions (namely that the subgroup $H$ must be normal), we may define the Quotient group, a very important concept; see the page on "left cosets partition the parent group" for a glance at why this is useful. [todo: there must be a less clumsy way to do it]

Additionally, there is a key theorem whose usual proof considers cosets (Lagrange's theorem) which strongly restricts the possible sizes of subgroups of $G$, and which itself is enough to classify all the groups of order $p$ for $p$ prime. Lagrange's theorem also has very common applications in [-number_theory], in the form of the [fermat_euler_theorem Fermat-Euler theorem].